The Properties And Applications Of Several Classes Of Generalized Convex Functions And Convexification, Concavification Method For Monotone Optimization

In this thesis, the properties of several classes of generalized convex functions and their applications in optimization problems such as extremum problems and dual problems etc. and a convexification, concavification method of monotone optimization problem are resarched.The first kind of generalized convex function is prequasi-invex and semistrictly prequas-invex funtions. Convex and generalized convex functions play a central role in optimization theories. Prequas-invex is an important generalized convex fuction, it is a generalization of quasi-convex function and invex function. So the study of prequasi-invex functions has some theoretical and practical singnificance. Professor Yang had obtained the necessary and sufficient conditions of prequasi-invex functions under condition of lower semicontinuity and Condition D. In this thesis, we obtained many equivalent conditions of prequasi-invex functions under much weaker conditions by applying nearly convexity of sets, that is to say we can obtain the prequasi-invexity in terms of the closeness of sets. Yang established many good properties of prequasi-invex functions and semistrictly prequasi-invex functions. In this thesis we obtained a sufficient condition and other properties of semistrictly prequasi-invex functions by weaking the conditions.The second generalized convex function we study in this thesis is B-preinvex functions. Bector and Singh introduced the B-vex functions by relaxing the defi-nition of convexity of a function. Sujea introduced the B-preinvex functions, thus united the B-vex functions and preinvex functions. In this thesis, we discuss other sufficient conditions and new properties. Futhermore, the sufficient optimality conditions and Mond-Weir type weak ang strong duality results are obtained for a nonlinear programming.(v, F, p, 0)-convex functions is the third functions we considered in this thesis. In recent years, optimality and duality for muli-objective fractional programs have been studed by many authors. Bector et al. derived Fritz John and Karush-Kuhn-Tucker necessary and sufficient optimality condition for a class of non-differentiable convex multi-objective fractional programming problems, also establised the duality results. Produ introduced the concept of (F, p)-convexity, an extension of F—convexity. Liu obtained necessary and sufficient conditions and derived duality theorems for a class of nonsmooth multiobjective fractional programming problems involving (F, p)-convex functions. KuK et al. denned the concept of (v, p)-invexity for vectorvalued functions, and they proved the generalized Karush-Kuhn-Tucker necessary and sufficient optimality theorem, weak and strong duality for nonsmooth multiobjective programs under the (v, p)-invexity assumptions. Bector et al. and Xu gave a mixed type duality for fractional programming, established some sufficient optimality conditions and obtained various duality results between the mixed duality problem and primal problem. In this thesis, we introduced a class of functions called (v, F, p, 0)convex functions which involvs the (F, p)convex functions and (v, p)-invex functions as special cases. And we establish generalized Karush-Kuhn-Tucker necessary and sufficient optimality conditions and introduce the mixed duality problem (MFXD) of non-smooth multi-objective fractional programming problems (MFP), in which functions are locally Lipschitz, obtain various duality results between themixed duality problem (MFXD) and primal problems (MFP) under the assumptions of (v, F, p, #)-convexity.The monotone optimization is a global optimization problem in which objective function and constrained functons are all monotone. This thesis proposes a new convexification or concavification transformation method to convert a monotone functioin into a convex or concave function. Then the monotone optimization problem can be converted into an equivalent concave minimization problem or reverse convex programming problem or canoical D.C. programming problem.